To solve the DM puzzle, an alternative to WIMP models is represented by the scenarios based on long lived super-heavy relics, that can be produced by several mechanisms taking place during the inflationary phase or just after, in the re-heating phase (see [1]}, [2]}, [3]}, [4]}, [5]} and references therein). Once created in the early universe, the abundance of the long-lived super-heavy particles can evolve to match the DM density observed today, the so-called Super Heavy Dark Matter (SHDM). This conclusion can be drawn under three general hypotheses: (i) SHDM in the early universe never reaches local thermal equilibrium; (ii) SHDM particles have mass \(M_X\)  of the order of the inflaton mass or higher; and (iii) SHDM particles are long-living particles with a lifetime exceeding the age of the universe, \(\tau _X\gg t_0\) .
